The following short notes provide a short recap on elementary concepts of module theory, which I will use throughout the lectures.

I assume you should have come across this material in a lecture on Rings and Modules / Commutative Algebra / Algebraic Geometry / ... If it is not the case, do not hesitate to let me know before the start of the school.

• Modules and algebras: basics
• The chain conditions, Jordan-Hölder, composition series
• The Jacobson radical
• Simplicity, indecomposability, semisimplicity

I have invested some time in producing a pdf version of my blackboard talks containing some general explanations and motivation about each topic treated. A first draft is available here:

• Lecture Notes (Version 0)

You can help me improve them! Please feel free to e-mail me if you find typos or have suggestions for improvements.
If you'd like a handout of my beamer presentations as well, just drop me an e-mail.

The exercises are to be found in the lecture notes. They are placed precisely where the relevant material to solve them has been presented.

Fur further reading on the topics treated in this mini-course, I recommend the following textbooks and surveys:

Notes from Summer School Lectures.

• [Bro92] M. Broué, Equivalences of blocks of group algebras. [MathSciNet]
• [Kes07] R. Kessar, Introducton to block theory. [MathSciNet]
• [Kue18] B. Külshammer, Basic local representation theory. [MathSciNet]
• [HKK10] G. Hiss, R. Kessar, B. Külshammer, An Introduction to the Representation Theory of Finite Groups [Unpublished]
• [Samb16] B. Sambale, Determination of block invariants [Unpublished]

A nice, smooth and complete introduction to representations of finite groups for beginners can be found in Peter Webb's book, which I highly recommend:

• [Web16] P. Webb, A course in finite group representation theory. [MathSciNet]

Further Books.

• [Alp86] J. L. Alperin, Local representation theory. [MathSciNet]
• [Ben98] D. J. Benson, Representations and cohomology I. [MathSciNet]
• [Cra19] D. Craven, Representation theory of finite groups: a guidebook. [MathSciNet]
• [CR81] C. Curtis and I. Reiner, Methods of representation theory. Vol. I. [MathSciNet]
• [Dor72] L. L. Dornhoff, Group representation theory. Part B: Modular representation theory. [MathSciNet]
• [EH18] K. Erdmann and T. Holm, Algebras and representation theory. [MathSciNet]
• [Lin18] M. Linckelmann, The block theory of finite group algebras. Vol. I and II.. [MathSciNet]
• [LP10] K. Lux and H. Pahlings, Representations of Groups, A Computational Approach. [MathSciNet]
• [NT89] H. Nagao and Y. Tsushima, Representations of finite groups. [MathSciNet]
• [Nav98] G. Navarro, Characters and blocks of finite groups. [MathSciNet]